The Eulerian numbers can D.I.E

TL;DR

This paper explores two alternating sign formulas for Eulerian numbers, demonstrating proofs via the D.I.E. involution method, and discusses their broader mathematical implications.

Contribution

It introduces new proofs for Eulerian number formulas using the D.I.E. method and generalizes these approaches to other mathematical contexts.

Findings

Two distinct sign formulas for Eulerian numbers are proved.

The D.I.E. involution technique is effectively applied to these proofs.

The methods suggest broader applications in combinatorics and related fields.

Abstract

The Eulerian numbers form a triangular array with many interesting properties. The numbers arise from various combinatorial and probabilistic interpretations, and have been studied in a variety of mathematical contexts. In this article we examine two distinct alternating sign formulas for the Eulerian numbers and show how they can be proved using a sign-reversing involution technique described by Benjamin and Quinn known as the ``D.I.E.'' method. Each of these arguments lends itself to a broad generalization, shedding light on different parts of mathematics.